Internal problem ID [11223]
Book: An elementary treatise on differential equations by Abraham Cohen. DC heath publishers.
1906
Section: Chapter IV, differential equations of the first order and higher degree than the first.
Article 26. Equations solvable for \(x\). Page 55
Problem number: Ex 2.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {a^{2} y {y^{\prime }}^{2}-2 x y^{\prime }+y=0} \]
✓ Solution by Maple
Time used: 0.515 (sec). Leaf size: 65
dsolve(a^2*y(x)*diff(y(x),x)^2-2*x*diff(y(x),x)+y(x)=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) = -\frac {x}{a} y \left (x \right ) = \frac {x}{a} y \left (x \right ) = 0 y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (\tanh \left (-\textit {\_Z} +c_{1} -\ln \left (x \right )\right )^{2} {\mathrm e}^{2 \textit {\_Z}} a^{2}-\tanh \left (-\textit {\_Z} +c_{1} -\ln \left (x \right )\right )^{2}+1\right )} x \end{align*}
✓ Solution by Mathematica
Time used: 30.099 (sec). Leaf size: 244
DSolve[a^2*y[x]*(y'[x])^2-2*x*y'[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )-8 i x}}{4 a} y(x)\to \frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )-8 i x}}{4 a} y(x)\to -\frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )+8 i x}}{4 a} y(x)\to \frac {\left (\cosh \left (\frac {a^2 c_1}{2}\right )+\sinh \left (\frac {a^2 c_1}{2}\right )\right ) \sqrt {\cosh \left (a^2 c_1\right )+\sinh \left (a^2 c_1\right )+8 i x}}{4 a} y(x)\to -\frac {x}{a} y(x)\to \frac {x}{a} \end{align*}